The Karpelevic region revisited

We consider the Karpelevic region Theta(n) subset of C consisting of all eigenvalues of all stochastic matrices of order Theta(n). We provide an alternative characterisation of Theta(n) that sharpens the original description given by Karpelevic. In particular, for each theta is an element of [0, 2 pi), we identify the point on the boundary of Theta(n) with argument theta. We further prove that if n is an element of N with n >= 2, and t is an element of Theta(n), then t is a subdominant eigenvalue of some stochastic matrix of order n. (C) 2020 Elsevier Inc. All rights reserved.